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MOTION                                     95
(c)  The magnitude of the total acceleration is given by
I       __
the relation                  / = \/ v~ +  
(d)  The total acceleration is directed towards the concave side of the path and makes an angle with the tangent which is defined by
(e) When the path is straight, that is, when p = oo , the normal acceleration is nil; therefore in this case the total acceleration is identical with the tangential acceleration.
(/) When the path is circular and the speed constant, then p = r, the radius of the circle, and v = 0; therefore
f-f -     v*
J  ~Jn-   -  - '
90. Radial and Transverse Components of Acceleration.  Let P (Fig. 58) be any point of the path at which the acceleration of the particle is to be considered. Take two neighboring points PI and P2, and let YI and v2 be the velocities at these points. Then the change in the radial velocity in going from Pi to P2 is obtained by subtracting the radial component of Vi from that of v2. Replace YI and v2 by their components along and at right angles to TI and r2, respectively, and denote these components by vri, vPl and vra, vpa; then it will be seen from the figure that
(vrz cos 2 - vPz sin 62) - (vri cos ci + vPi sin ei)
is the total change in the radial velocity. Therefore the radial component of the acceleration is
, = ijmjt r^os 2 - PP. sin e2 - vr, cos i - t;gl sin <{]
*=o L                                    t                                    J
where t is the time taken by the particle to go from Pi to P2. But as the points Pi and P2 approach P as a limit, the following substitutions become permissible.